A THESIS SUBMITTED TO DEPARTMENT OF MATHEMATICS COLLEGE OF NATURAL SCIENCE SCHOOL OF GRADUATE STUDIES ARBA MINCH UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE IN MATHEMATICS WITH SPECIALIZATION IN DIFFERENTIAL EQUATION

Abstract

The Dirichlet problem in the disk is solved two ways. First, we use the real and imaginary part of z n together with Fourier series. A Fourier analysis began as an attempt to approximate periodic functions with in nite summations of trigonometric polynomials. For certain functions, these sums, known as Fourier series, converge exactly to the original function. Then next we construct Poisson kernel by using Fourier series along other trigonometric polynomials, in particular that they are sum of holomorphic and anti-holomorphic functions.